Comparison geometry for an extension of Ricci tensor

For a complete Riemannian manifold $M$ with an (1,1)-elliptic Codazzi self-adjoint tensor field $A$ on it, we use the divergence type operator ${L_A}(u): = div(A\nabla u)$ and an extension of the Ricci tensor to extend some major comparison theorems in Riemannian geometry. In fact we extend theorems like mean curvature comparison theorem, Bishop-Gromov volume comparison theorem, Cheeger-Gromoll splitting theorem and some of their famous topological consequences. Also we get an upper bound for the end of manifolds by restrictions on the extended Ricci tensor. The results can be applicable for some kind of Riemannian hypersurfaces when the ambient manifold is Riemannian or Lorentzian with constant sectional curvature.